## Numerical evidence for the equivariant Birch and Swinnerton-Dyer conjecture.(English)Zbl 1277.11072

Summary: Let $$E/ \mathbb Q$$ be an elliptic curve and $$K/ \mathbb Q$$ a finite Galois extension with group $$G$$. We write $$E_K$$ for the base change of $$E$$ and consider the equivariant Tamagawa number conjecture for the pair $$(h^1(E_K)(1),\mathbb Z[G])$$. This conjecture is an equivariant refinement of the Birch and Swinnerton-Dyer conjecture for $$E/K$$. For almost all primes $$l$$, we derive an explicit formulation of the conjecture that makes it amenable to numerical verifications. We use this to provide convincing numerical evidence in favor of the conjecture.

### MSC:

 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11G05 Elliptic curves over global fields

Zbl 1277.11071
Full Text:

### References:

 [1] Bley [Bley 10] W., Math. Comp. [2] Bley [Bley and Burns 03] W., Proc. London Math. Soc. 87 pp 545– (2003) · Zbl 1056.11071 [3] Bley [Bley and Johnston 08] W., J. Algebra (Computational Section) 320 pp 836– (2008) [4] Bley [Bley and Johnston 11] W., Math. Comp. 80 pp 2411– (2011) · Zbl 1273.16016 [5] Bley [Bley and Wilson 09] W., London Math. Soc. J. Comp. and Math. 12 pp 166– (2009) · Zbl 0016.39203 [6] Bloch [Bloch and Kato 90] S., The Grothendieckfestschrift, vol. 1, Progress in Math. 86 pp 333– (1990) [7] Bouganis [Bouganis and Dokchitser 07] T., Math. Proc. Cambridge Philos. Soc. 142 pp 193– (2007) · Zbl 1214.11080 [8] Breuning [Breuning 04] M., Math. Comp. 73 pp 881– (2004) · Zbl 1041.11073 [9] Breuning [Breuning and Burns 05] M., Homology, Homotopy and Applications 7 pp 11– (2005) · Zbl 1085.18011 [10] Breuning [Breuning and Burns 07] M., Compositio Math. 143 pp 1427– (2007) [11] Burns [Burns 04] D., Number Theory, CRM Proceedings and Lecture Notes 36 pp 35– (2004) [12] Burns [Burns 09] D., Pure App. Math. Q. 6 pp 83– (2010) · Zbl 1227.11118 [13] Burns [Burns and Flach 96] D., Math. Ann. 305 pp 65– (1996) · Zbl 0867.11081 [14] Burns [Burns and Flach 01] D., Documenta Math. 6 pp 501– (2001) [15] Curtis [Curtis and Reiner 87] C., Methods of Representation Theory, vols. I and II (1981) [16] Darmon [Darmon 06] H., International Congress of Mathematicians, vol. II pp 313– (2006) [17] Dokchitser [Dokchitser 04] T., Experiment. Math. 13 pp 137– (2004) [18] DOI: 10.1080/10586458.2010.10129069 · Zbl 1221.11148 [19] Flach [Flach 04] M., Stark’s Conjecture: Recent Progress and New Directions, Contemp. Math. 358 pp 79– (2004) [20] Flach [Flach 09] M., Pure and Applied Mathematics Quarterly 5 pp 255– (2009) · Zbl 1179.11036 [21] Fröhlich [Fröhlich 89] A., J. Reine Angew. Math 397 pp 42– (1989) · Zbl 0693.12012 [22] Gross [Gross and Zagier 86] B. H., Inventiones Mathematicae 84 pp 225– (1986) · Zbl 0608.14019 [23] Grothendieck [Grothendieck 72] A., Groupes de Monodromie en Géométrie Algébrique (SGA 7 I), Lecture Notes in Math. 288 (1972) [24] Kings [Kings 09] G., ”An Introduction to the Equivariant Tamagawa Number Conjecture: The Relation to the Birch–Swinnerton-Dyer Conjecture.” Preprint Nr. 26/2009 (2009) [25] Kolyvagin [Kolyvagin 90] V. A., The Grothendieck Festschrift, Progr. in Math. 87 pp 435– (1990) [26] Kolyvagin [Kolyvagin and Logachev 90] V. A., Leningrad Math. J. 1 pp 1229– (1990) [27] Kolyvagin [Kolyvagin and Logachev 92] V. A., USSR Izvestiya 39 pp 829– (1992) · Zbl 0791.14019 [28] Mazur [Mazur and Tate 87] B., Duke Math. J. 54 pp 711– (1987) · Zbl 0636.14004 [29] Navilarekallu [Navilarekallu 88] T., International Mathematics Research Notices pp 33– (2008) [30] Niziol [Niziol 93] W., Duke Math. J. 71 pp 747– (1993) · Zbl 0803.14008 [31] DOI: 10.1007/BF01391466 · Zbl 0363.10019 [32] DOI: 10.1215/S0012-7094-78-04529-5 · Zbl 0394.10015 [33] Silverman [Silverman 86] J. H., The Arithmetic of Elliptic Curves. (1986) [34] Swan [Swan 68] R. G., Algebraic K-Theory, Lecture Notes in Mathematics 76 (1968) [35] Venjakob [Venjakob 07] O., L-Functions and Galois Representations, London Math. Soc. Lecture Note Ser. 320 pp 333– (2007) [36] Zhang [Zhang 01] S., Annals of Math. 153 pp 27– (2001) · Zbl 1036.11029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.